SINGULARITY THEORY AND FORCED SYMMETRY , and Angela Maria

53 Pages·2010·0.53 MB·English

Publ.Mat.54(2010), 263–315
SINGULARITY THEORY AND FORCED SYMMETRY
BREAKING IN EQUATIONS
Jacques-Elie Furter , Maria Aparecida Soares Ruas , and
∗ †
Angela Maria Sitta
‡
Abstract
Atheoryofbifurcationequivalenceforforcedsymmetrybreaking
bifurcationproblemsisdeveloped. Weclassify(O(2),1)problems
ofcorank2oflowcodimensionanddiscussexamplesofbifurcation
problemsleadingtosuchsymmetrybreaking.
1. Introduction
The importance of symmetries in understanding and influencing bi-
furcation problems has been recognised basically since the beginning of
bifurcation theory. Symmetries influence bifurcation problems via two
broad classes of mechanisms:
spontaneous symmetry breaking,
•
forced (or induced) symmetry breaking.
•
Spontaneous symmetry breaking occurs when the symmetry of the
equationsisconstant whereassolutionsbifurcateandlose(orgain)inter-
nalsymmetry as the parametersvary. Fromthe many problems leading
to bifurcation equations with spontaneous symmetry breaking,we men-
tionforlaterreference,theanalysisofthebehaviourofanhomogeneous,
isotropicelastic cube under uniform traction(its symmetry groupis the
permutation group S , see [20]) or the bifurcation of periodic orbits in
3
autonomous systems (its symmetry group for the non-resonant degen-
erate Hopf bifurcation is the rotation group S1, see [19]). A singularity
theoryapproachtothelocalstudyofthebifurcationdiagrams,basedon
2000 Mathematics Subject Classification. 37G40, 37G05,58K40,58K70.
Keywords. Singularitytheory,bifurcation,forcedsymmetrybreaking.
∗PartiallysupportedbyCCInt-USP andFUNDUNESP-UNESP.
†PartiallysupportedbyCNPq,FAPESPandCAPES.
‡PartiallysupportedbyFAPESPandCAPES.
264 J.-E. Furter, M. A. Soares Ruas, A. M. Sitta
equivariant contact equivalence, has been developed by Golubitsky and
Schaeffer ([19], [20]). The original set-up is as follows.
1.1. Singularity Theory for Spontaneous Symmetry Breaking.
First, because such analysis is local, it is sensible to consider germs
at the origin to be able to state results that will persist on any neigh-
bourhoodof the origin. A germ of a function arounda point x is an
0
equivalence class when two functions are identified if they coincide in a
neighbourhoodof that point. We use the notation f: (Rn,x ) Rm to
0
→
denote the germ of f around x . To make sense of equations like f =0,
0
when f: (Rn,x ) Rm, we define germs of sets, or germs of vari-
0
→
eties,atx usingthesameidentificationprocessasforgermoffunctions
0
at x . Let Γ be a compact group acting on Rn and Rm, we say that
0
f: (Rn,0) (Rm,0) is Γ-equivariant, resp. Γ-invariant, if f(γx) =
γf(x),resp→.f(γx)=f(x), x (Rn,0), γ Γ. Abifurcation germ
with l parameters is a germ∀ f∈: (Rn+l,0)∀ ∈(Rm,0). Its bifurcation
diagram is its zero-setf 1(0). LetΓ be a→compactgroupacting onRn
−
and Rm, trivially on parameters, two Γ-equivariant bifurcation germs
(withonedistinguishedbifurcationparameter)f,g: (Rn+1,0) (Rm,0)
→
arebifurcation,or Γ,equivalentifthereexistΓ-equivariantchanges
Kλ
of coordinates (T,X,L) such that
(1.1) f(x,λ)=T(x,λ)g(X(x,λ),L(λ)),
where T(x,λ) is an invertible matrix and (X,L) a local diffeomorphism
around the origin. Note the special role of λ which means that λ-slices
of the bifurcation diagrams of f and g are diffeomorphic via (X,L).
The use of singularity theory leads to a systematic understanding and
systematic calculation of the relative roles of the different terms of a
bifurcationgerm, and, via the notion of miniversalunfolding, of its per-
turbations. The theory based on (1.1) established itself for the local
study of bifurcation germs, giving efficient algebraic calculations to es-
tablish classificationsof normal forms, special polynomialmembers of
the orbits under bifurcation equivalence, and their miniversal defor-
mations, perturbations of the normal forms with the minimal number
of parameters necessary to represent all possible perturbations modulo
changes of co-ordinates. Given a bifurcation germ f, the key algebraic
ingredient of the theory is the extended tangent space of f, denoted
by Γ(f). It has a so-calledmixed module structure because it is the
TeKλ
sum of a module generated over the ring of Γ-invariant germs in (x,λ)
and of a module generated over germs in λ only. An important number
Forced Symmetry Breaking 265
associatedwith f is the codimensionof Γ(f) as a vectorsubspace of
thespaceofΓ-equivariantmaps. ItiscaTlleeKdλthe KΓ-codimensionoff,
λ
denoted by Γ-cod(f). When Γ-cod(f) is finite, f is Γ-equivalent
Kλ Kλ Kλ
to a polynomial normal form and has a miniversal unfolding with Γ-
Kλ
cod(f) parameters, formed using a basis a complement of Γ(f) in
TeKλ
the space of Γ-equivariant bifurcation maps, the normal space of f,
denoted by Γ(f).
NeKλ
1.2. Forced Symmetry Breaking.
Inthesecondmechanism,thatofforcedsymmetrybreaking,thesym-
metriesofthe equationschange when someparametersareswitchedon.
For example, in the elastic cube problem, the traction load distribution
could change first to remain equal only on two opposite faces before
losing any symmetry with a second additional parameter. The forced
symmetry breaking is then S Z 1 (with one, then two additional
3 2
→ →
parameters coming into play, see [13]).
Therearemanyotherbucklingproblemsinelasticityexhibitingforced
symmetry breaking because there is often an interaction between the
internal symmetries of the material, the geometry of the object and of
the externally applied forces. An example is the work of Pierce [31] on
the bifurcation of straight circular rods subject to an axially symmetric
compressive load and perturbations by additional loads breaking the
axial symmetry. In previous works on prismatic rods, like in [3], the
typical diagrams are to be found in the miniversal unfolding of the non
degeneratedoublecuspbecausethesymmetrygroupisfinite. Whenthe
rodiscircularthedoublecuspisdegeneratebecausetheactingsymmetry
groupisnowO(2)andsotheproblemhasnominiversalunfoldingwithin
the classical theory.
Similarly, the forcing of an autonomous equation u¨+q(u,λ) = 0 by
either a T-periodic function p, like in the following model problem
(1.2) u¨+q(u,λ)+µp(t)=0,
ornonlinearboundaryconditionsofthetype(g ,g are(nonlocal)non
1 2
linear functions of u):
u(0) u(1)=µg (u,λ,µ), u˙(0) u˙(1)=µg (u,λ,µ),
1 2
− −
gives rise to a bifurcation problem with the forced symmetry break-
ing O(2) 1 near the values of λ where the linear mode of the au-
→
tonomous part is of period-T also. We can extend the symmetry break-
ing bifurcation for any pair of compact subgroups ∆ Γ O(2), by
⊂ ⊂
considering a generalisation of the differential equation (1.2) with time
266 J.-E. Furter, M. A. Soares Ruas, A. M. Sitta
periodic terms g (nonlinear) and p with rationally dependent periods
(see [17] for a classical analysis).
1.3. Main Results.
A special version of bifurcation equivalence, stronger than (1.1), is
called for to classify systematically the bifurcation germs and their un-
foldings arising in forced symmetry breaking problems.
1.3.1. Section 2: abstract theory.
In Section 2, we present a general theory of unfoldings, finite de-
terminacy and the recognition problem for forced symmetry breaking
bifurcation germs of the type
f: (Rn R R,0) (Rn,0), (x,λ,µ) f(x,λ,µ),
× × → 7→
where f is Γ-equivariant when µ = 0 and ∆-equivariant when µ = 0,
6
for ∆ a closed subgroup of Γ. The essential part of the theory was ad-
vanced in [18] but not much action was taken on it, the main theory
drifting instead towards the very rich field of spontaneous symmetry
breaking. But, fundamentally, Damon in [8] adapted his general frame-
worktoclearthewayfortheabstracttheorytoworksuccessfullyinthat
case. So, we may write
(1.3) f(x,λ,µ)=f (x,λ)+µf (x,λ,µ)
1 2
with f Γ-equivariant and f ∆-equivariant. A first approach to study
1 2
such problems, is to view f as a perturbation of f in the ∆ -theory,
1 K(λ,µ)
the bifurcation equivalence of ∆-equivariant maps with two bifurcation
parameters (λ,µ). In general that approach fails. For instance, when
Γ and ∆ have different dimensions, the ∆ -codimension of f is not
K(λ,µ) 1
finite (even when Γ is finite, it may fail). So, the group of change of
coordinates Γ,∆ we define in Section 2.3 will have the property that
K(λ,µ)
(T,X,L)isΓ-equivariantwhenµ=0,butonly∆-equivariantwhenµ=
6
0. This(Γ,∆)-equivariantstructureofthegroupofcontactequivalences
will be transported to the tangent spaces (see Section 2.3.4). They are
modules over Γ-invariant functions when µ = 0 and over ∆-invariant
functions when µ = 0. This is unusual but, in his general framework,
6
Damon did define in [8] the necessary extended concepts to deal with
the new situation.
ThefirstpointistomakesurethataversionofthePreparationTheo-
remapplies. Asaconsequence,wecanworkwiththealgebraicstructure
of the tangent spaces. For the usual theory of Γ-equivalence, the ring
Kλ
of invariant functions has a structure of differentiable DA-algebra (see
Forced Symmetry Breaking 267
Section 2.2), and so the Preparation Theorem holds true. When Γ is
a continuous Lie group and ∆ is finite, our rings of invariant functions
will not be DA-algebras, but Damon showed that the main properties
of DA-algebras can be extended to this situation of so-called extended
DA-algebras. InCorollary2.2wegiveanexplicitcriterionweusetotest
if the rings of invariants under consideration are actually DA-algebras.
In a second step, we look into the structure of our group of contact
equivalences Γ,∆ andthe tangentspaces. InSection 2.3 we show that
K(λ,µ)
Γ,∆ decomposesintothreesubgroups: Kˆ(Γ,∆), Mˆ(Γ,∆)andSˆ(Γ,∆).
K(λ,µ)
When Γ is continuous and ∆ is finite, for instance, each subgroup add
a non trivial contribution to the extended tangent space of Γ,∆ . In
K(λ,µ)
Section 2.3.5 we show that Γ,∆ is a geometric subgroup of (con-
K(λ,µ) K
tactequivalences), hence it satisfies the abstracttheorems of Damon[8]
about miniversal unfoldings and finite determinacy. We show in The-
orem 2.7 that the explicit description of Γ,∆ in [18] corresponds to
K(λ,µ)
the best possible situation, because its extended tangent space contains
the extended tangent space of any other geometric subgroupof fixing
K
globally the Γ-equivariant maps when µ=0.
Finally, we discuss topological Γ,∆ -equivalence where the changes
K(λ,µ)
ofco-ordinates(T,X,L)areonlycontinuous,notsmooth,germs. InSec-
tion 2.5, we summarise the results of Damon [10], [11] we need. Topo-
logical equivalence is more efficient in our context, because it preserves
the topological properties of the bifurcation germs and their (smooth)
miniversal unfoldings. Our smooth normal forms and their smooth
miniversal unfoldings have many moduli (parameters without topologi-
cal significance). They appear because the geometry of the bifurcation
diagrams is intricate (see Figure 4). As most two parameter situations,
the Γ,∆ -equivalence preservesthe respective positionof the regionsof
K(λ,µ)
theparameterplanewithadifferentzerostructure. Inourcasetheonly
one dimensional slice to be preserved is f1−1(0), when µ=0.
1.3.2. Section3: (O(2),1)-symmetrybreakingclassification.
As an example, we classify (O(2),1)-symmetry breaking problems.
Although we stop at topological codimension 1, we still have germs of
highsmoothcodimensionwithanintricateregionstructureforthezero-
set (see Figure 4 in Section 4, for instance) because the normal forms
can have many moduli parameters (parameters invariant under smooth
Γ,∆ -equivalence). In Theorem 3.4 (about topological Γ,∆ -equiva-
K(λ,µ) K(λ,µ)
lence)weshowthatthereare3normalformsuptotopologicalcodimen-
268 J.-E. Furter, M. A. Soares Ruas, A. M. Sitta
sion 1. They all satisfy f (0,0,0) = 0, and their smooth codimension
2
6
vary from 0 to 4 as shown in Theorem 3.5 (about smooth Γ,∆ -equiv-
K(λ,µ)
alence).
We believe that we have a complete list up to topological codimen-
sion 2, adding 4 more normal forms, but we cannot fully prove it at
present, so we simply mention that list as a remark following Theo-
rem 3.4. Interestingly, one normal form has f (0,0,0) = 0. Note that
2
the highest smooth codimension of those germs can reach is 12. As a
consequence of our computations of the tangent spaces, the list of the
(Z ,1)-symmetry breaking germs f in one dimension, that is, f(y,λ)=
2
f (y2,λ)y+µf (y,λ,µ), isbasicallythe sameasthe(O(2),1)-classifica-
1 2
tionlistwhenf (0,0,0)=0. Nevertheless,thestabilitypropertiesofthe
2
6
solutions are different (see [12]). In the corank two case, the symmetry
breaking term selects only one stable solution from the O(2)-orbit. In
theonedimensionalcase,theobviousobstructionsfromtheunstableso-
lutionsmeanthatthestabilitypropertiesofthepairsofsolutionsremain
unchanged.
1.3.3. Section 4: (O(2),1)-symmetry breaking bifurcation dia-
grams.
In this section, we look at the bifurcation diagrams for the (O(2),1)-
symmetry breaking bifurcation germs f we classified in Theorem 3.4,
cases I , II and III. In Figure 2 of Section 4.2.1, we give a simple illus-
0
tration on a potential dynamics of what happen when orbits of steady
states are destroyed by the perturbation of the miniversal unfolding of
caseI . Moregenerally,wedescribetheregionsofparameterspace(λ,µ)
0
where the zero set structure of f is invariant, including the stability of
the solution with respect to the sign of the eigenvalues of the linearisa-
tion of f. They are portrayed in Figures 1, 3 and 4, respectively. For
bifurcation equivalent germs, those regions are diffeomorphic but the
only one dimensional slice preserved is µ=0.
1.3.4. Section 5: examples.
In the finalsection we describe examples leading to bifurcation equa-
tions satisfyingour framework. Our results arereadily applicableto the
bifurcation of period-2π solutions of
u¨+u+uq(u,λ)+µp(t,u,u˙,λ,µ)=0
where p is 2π-periodic in t and q(0,0) = 0. This example is technically
easy to manipulate. As it is well-known, when π p(t,0)eitdt = 0,
| π | 6
the symmetry breaking term selects a pair of soluRt−ions (that is, selects
Forced Symmetry Breaking 269
a phase shift) from the O(2)-orbit, keeping only one solution possibly
stable. In that case the problem reduces to the corank one symmetry
breaking (Z ,1).
2
Another problemis concernedwith the problemof contactresistance
in homogeneous metal rings obtained from welding the end points of a
piece of metal. Usually the joining is not perfect and give rise to some
‘contact resistance’ to heat conduction. This resistance is characterised
by a continuous gradient of the temperature across the join (the mate-
rial is the same across the join on the left and the right) and a loss of
temperature at first approximation proportional to its gradient. More
explicitly,ifthepieceofmetalis[0,1],thesimplestboundaryconditions
are
(1.4) u(0) u(1)=µu˙(1) and u˙(1)=u˙(0).
−
The boundary conditions are periodic if µ=0 and have no symmetries
when µ = 0. Combined with an autonomous non linear heat equation
6
theygiverisetoaforced(O(2),1)-symmetrybreaking(see[34]and[29]).
Another type of problems are reaction-diffusion equations. Let u be
the density of a population living in the unit disk. Suppose u satisfies a
parametrised semilinear parabolic equation
(1.5) u =∆u+g(u,u¯,λ)+µp(x,y,u,u¯,λ,µ)
t
subject to non-flux boundary conditions ∂u =0 on the unit circle. The
∂n
termu¯representsanonlocalcontributionoftheaverageddensityu¯(t)=
1 u(s,t)ds. Under the hypotheses of Section 4,(1.5) has a constant
Ω Ω
s|te|aRdystatewhichloosesstabilitywitha2-Dkernelforthelinearisation.
For general p, the resulting bifurcation equation fits our framework. As
anexampleof(Z ,1)-symmetrybreaking,takeagenericthinrectangular
2
plate under a uniform compression along its boundary. Apply a generic
normal load distribution and use the von K´arm´an approximation for
the buckling equations (see [36]). The unperturbed equations have a
Z -symmetry, destroyed by the perturbing load distribution.
2
1.4. Comments and Some Related Work.
1.4.1. Variational problems.
Bifurcation equations in elasticity are often the gradients of some
parametrisedfunctional (see [31]). Bifurcation equivalences do not pre-
serve in general the gradient structure, but they still induce an equiva-
lence relation on the set of gradient bifurcation germs and their unfold-
ings. Atheoryusingapathformulationhasbeendevelopedin[2],butit
270 J.-E. Furter, M. A. Soares Ruas, A. M. Sitta
is not clear how to extend those ideas to the (O(2),1)-symmetry break-
ing,butnomorepowerfultheoryisneededhere. Ournormalforms(with
caseI forsomevaluesofitsmoduli)andtheirminiversalunfoldingsare
2
already gradients. If any other gradient bifurcation germ satisfies the
recognition conditions, then it is bifurcation equivalent to the represen-
tative normal form, which is a gradient. Moreover, the same applies to
the miniversal unfoldings that are also gradients. This means that the
classification of gradient problems mirrors the general classification.
1.4.2. Symmetries on parameters.
Forced symmetry breaking arises also when some symmetries are re-
tained by the parameters(see [15]). A typical example is when we have
a non trivial ‘diagonal’ action of Γ on the space of (x,λ), γ (γx,γλ).
7→
Inthatcasethetheoryinthispaperisnotnecessary. Thegeneralstruc-
ture of the problem remains ‘classical’ because the invariants form a
DA-algebra (see [15]). But, if we combine both ideas: some symme-
try is lost, some is retained by the parameters when they are switched
on, we get a theory which is still contained into the abstract frame-
work we describe here because the invariants still form a DA-algebra.
When combined with the symmetry breaking terms, we have a struc-
ture of extended DA-algebras for the invariants. More explicitly, the
problemsrepresentedby the functions f(x,λ,µ)=f (x,λ)+f (x,λ,µ),
1 2
with f (γx,γλ) = γf (x,λ) and f (δx,δλ,δµ) = δf (x,λ,µ), fall into a
1 1 2 2
similar framework.
1.4.3. Bifurcations from orbits of solutions under perturba-
tions.
Somepreviousworkhavebeenconcernedwiththesymmetryofbifur-
cating solutions of (1.3). In the Γ-equivariant situation one can some-
times show that all bifurcating solutions have at least isotropy ∆ for
some subgroup ∆ Γ (see [36] for an abstract analysis). An example
⊂
of such analysis is the generalised Duffing’s equation (see [22], [23])
(1.6) u¨+u+uq(u,λ)+µp(t)=0
where q(0,0) = 0 and p is an even 2π-periodic function. One can then
show that all solutions of (1.6) are also even. Similar results hold when
q is an odd function and p is odd in time.
Systematic approaches can be traced back to Vanderbauwhede [35]
andLauterbachandRoberts[26],givingrisetoHouandGolubitsky[25]
(see Chapter 10 of [6] for a more recent discussion). The bifurcation
from Γ-orbits of solutions is studied when a ∆-equivariant perturbation
Forced Symmetry Breaking 271
isappliedtothe originalΓ-equivariantequations. Forinstance,asimple
example in tune with previous example is the equation u¨ + q(u,λ)+
µh(t,u,u˙)=0whenhis2π-periodicintandu¨+q(u)=0hasnontrivial
periodic orbits (see [36] and [4]). More generally, there has been more
recently interest about the effect of forced symmetry breaking on the
dynamics around some special solutions, invariant manifolds (Galante
and Rodrigues [17], Chillingworth and Lauterbach [5], Comanici [7],
Parker et al. [30] to mention but a few). In these papers, somewhat
differenttechniquesareused, linkedwith equivariantdifferentialgeome-
try. A complete unfolding of the underlying singularity near bifurcation
is difficult. However, in this paper, we are able to provide just such
analysis.
2. General Theory
In this section we present a generaltheory of unfoldings, finite deter-
minacy and the recognition problem for the bifurcation problems with
forcedsymmetrybreaking. Themainideasanddefinitionsweresketched
in[18] and[8]. Here we give a full, carefulandorganisedaccountof the
theory with some extended results. This needs many ingredients with
many definitions and results. The abstract theory of Damon [8] works
with modules over systems of rings that are DA-algebras. In general,
the ring of forced symmetry breaking invariant functions has only the
structureofanextended DA-algebra. InCorollary2.2wegiveacriterion
we use to test if an extended DA-algebra is actually a DA-algebra. In
that case we do not need any extension of the abstract theory. On the
contrary,likeforthe(O(2),1)-symmetrybreaking,anextendedtheoryis
required. InSection2.3,wediscussthestructureofthegroupofbifurca-
tionequivalence Γ,∆ wearegoingtouse. Itrequiresthecompositionof
K(λ,µ)
anyfinitestringofelementsofthreesubgroups. Onesubgroup,Kˆ(Γ,∆),
is natural, the two others, Mˆ(Γ,∆) and Sˆ(Γ,∆), are more surprising.
In Lemma 2.4 we show that the order of the elements is irrelevant: we
can always recombine any string as 3 elements, one in each subgroup.
The group Γ,∆ is a geometricsubgroupin the sense of [8] overan ad-
K(λ,µ)
equately ordered extended system of DA-algebras. To characterise the
tangent spaces we calculate the unusual contributions of Sˆ(Γ,∆) and
Mˆ(Γ,∆) in Lemmas 2.5 and 2.6, respectively. In Theorem 2.7 we show
that the tangent space of any other group of equivalence respecting the
forced symmetry breaking structure cannot be larger than the tangent
space of Γ,∆ , indicating that the choices of Sˆ(Γ,∆) and Mˆ(Γ,∆) are
K(λ,µ)
272 J.-E. Furter, M. A. Soares Ruas, A. M. Sitta
optimal. InSection2.4westatethemainresultswecandeducefromthe
abstract theory of [8] about the unfolding (Theorem 2.8) and determi-
nacy(Theorem2.9)theoriesfor Γ,∆ . Finally,mostnormalformshave
K(λ,µ)
moduli for the smooth equivalence and so we use the topological equiv-
alence theory of [10], [11] to regroup orbits of the smooth classification
into classes with equivalent topologicalbehaviour. The main results are
stated in Theorems 2.13 and 2.14.
2.1. Notation and Preliminary Definitions.
Thestatevariableisx (Rn,0)andthedistinguishedbifurcationpa-
rametersare(λ,µ) (R2,∈0). Thederivativesaredenotedby subscripts,
∈
f for ∂f,... and the superscripto denotes the value of any function at
x ∂x
theorigin,fo =f(0),fo =f (0),... . Let denotethe ringofsmooth
germs f: (Rn,0) R axnd x its maximaElxideal. For y Rm, let
x x,y
denote the -mo→dule of smMooth germs g: (Rn,0) Rm,∈and Ethe
x x,y
E → M
submodule of germs vanishing at the origin. When y is clear from the
context,wedenote by ~ and by ~ . Whenwewouldliketo
x,y x x,y x
E E M M
emphasiseonly the dimensionof the source we denote , by , ,
x y n m
E E E E
etc. To represent invariant and equivariant germs in terms of invariant
polynomials, it is convenient to use the following concept. Let X, Y, Z
be sets, f: X Y, g: Z Y, h: X Z be maps, f is the pullback
→ → →
of g by h, denoted by f =h g, if f(x)=g(h(x)), x X.
∗
∀ ∈
LetGL(n) be the groupofallinvertiblen n-realmatricesandO(n)
×
the n-dimensional orthogonal group. Let Γ be a compact Lie group
acting on Rn via an orthogonalrepresentationρ: Γ O(n). We denote
→
byΓ theconnectedcomponentoftheidentityinΓ,identifyγ withρ(γ),
0
γ Γ, and denote by γ the action on Rn induced by ρ. We denote by
∀ ∈
GL (n) the group M GL(n) : Mγ = γM, γ Γ of Γ-equivariant
Γ
{ ∈ ∀ ∈ }
matricesinGL(n)andby o(n)itsconnectedcomponentoftheidentity.
LΓ
2.1.1. Invariant functions.
Let Γ be the ring of smooth Γ-invariantgerms h: (Rn+1,0) R,
E(x,λ) →
h(γx,λ) = h(x,λ), γ Γ, and Γ its maximal ideal. Because
∀ ∈ M(x,λ)
Γ does not act on λ, there exists a finite set of Γ-invariant polynomials
u¯ (x) r such that any element h Γ can be written as the pull-
{ i }i=1 ∈E(x,λ)
back by u¯=(u¯ ,...,u¯ ,λ) of a function of u=(u ,...,u ) and λ, that
1 r 1 r
is, Γ = u¯ [33]. Similarly, for a closed subgroup ∆ Γ, we
E(x,λ) ∗E(u,λ) ⊂
define ∆ as the ring of ∆-invariant germs h: (Rn+2,0) R which
E(x,λ,µ) →
is the pullback by the ∆-invariant generators v¯ = (v¯ ,...,v¯ ,λ,µ) of a
1 s
germ of v = (v ,...,v ), λ and µ, that is, ∆ = v¯ . Finally,
1 s E(x,λ,µ) ∗E(v,λ,µ)